Completely regular and orthodox congruences on regular semigroups

Branka P. Alimpi\'c and Dragica N. Krgovi\'c

Abstract: Let $S$ be a regular semigroup and $E(S)$ the set of all idempotents of $S$. Let $\operatorname{Con}} S$ be the congruence lattice of $S$, and let $T$, $K$, $U$ and $V$ be equivalences on $\operatorname{Con}} S$ defined by $\rho T\xi \Leftrightarrow \tr\rho = \tr\xi$,\ $\rho K\xi \Leftrightarrow \ker \rho = \ker \xi$, $\rho U\xi \Leftrightarrow \rho \cap \leq = \xi\, \cap \leq$ and $V = U\cap K$, where $\tr\rho = \rho \mid_{E(S)}$,\enskip $\ker \rho = E(S)\rho$, and $\leq$ is the natural partial order on $E(S)$. It is known that $T$, $U$ and $V$ are complete congruences on $\operatorname{Con}} S$ and $T$-, $K$-, $U$- and $V$-classes are intervals $[\rho_T,\rho^T]$, $[\rho_K,\rho^K]$, $[\rho_U,\rho^U]$, and $[\rho_V,\rho^V]$, respectively ([13], [10], [9]). In this paper $U$-classes for which $\rho^U$ is a semilattice congruence, and $V$-classes for which $\rho^V$ is an inverse congruence are considered. It turns out that the union of all such $U$-classes is the lattice CR$\operatorname{Con}} S$ of all completely regular congruences on $S$, and the union of all such $V$-classes is the lattice O$\operatorname{Con}} S$ of all orthodox congruences on $S$. Also, some complete epimorphisms of the form $\rho \to \rho^U$ and $\rho \to \rho^V$ are obtained.